Ahmad Almulhem, KFUPM 2010

COE 202: Digital Logic DesignNumber Systems

Part 4

Dr. Ahmad AlmulhemEmail: ahmadsm AT kfupm

Phone: 860-7554Office: 22-324

Objectives

• Binary codes• Binary coded decimal (BCD)

• Other Decimal Codes

• Gray Code

• ASCII Code

• Error Detecting Code

Ahmad Almulhem, KFUPM 2010

Binary Codes

• A n-bit binary code is a binary string of 0s and 1s of size n.

• It can represent 2n different elements.• 4 elements can coded using 2 bits

• 8 elements can be coded using 3 bits

• Given the number of elements to be coded, there is a minimum number of bits, but no maximum !

Ahmad Almulhem, KFUPM 2010

Binary Coded Decimal (BCD)

• Human communicating with computers• Humans understand decimal• Computers understands binary

• Solution: Convert Decimal-Binary-Decimal• Need to store decimal numbers as binary

codes

Ahmad Almulhem, KFUPM 2010

Binary Coded Decimal (BCD)

• BCD Code uses 4 bits to represent the 10 decimal digits {0 to 9}• 6 BCD codes unused• The weights of the individual positions of the bits of a BCD code

are: 23=8, 22=4, 21=2, 20=1

Ahmad Almulhem, KFUPM 2010

Ahmad Almulhem, KFUPM 2010

BCD Addition

Other Decimal Codes• 4 bits = 16 different

codes

• Only 10 needed to represent the 10 decimal digits.

• Many possible codes!

• 2421 and excess-3 are self-complementing (9’s complement can be obtained by inverting bits)

Ahmad Almulhem, KFUPM 2010

src: Mano’s book

Gray Code

• Gray code represents decimal numbers 0 to 15 using 16 4-bit codes

• Gray codes of two adjacent decimal numbers differ by only one bit

• Example:• (5)10 = 0111

• (6)10 = 0101

• (7)10 = 0100

Ahmad Almulhem, KFUPM 2010

ASCII Character Code

• ASCII an abbreviation of “American Standard Code for Information Interchange”

• A 7-bit code (128 characters)• 94 printable, 34 non-printable (control)

• 2x26 English letters (A,…Z, a,…z)• 10 decimal digits (0,1,…9)• 32 Special Characters such as %, *, $, … etc.

• Usually stored as a byte (8 bits)• The extra bit is used for other purposes

Ahmad Almulhem, KFUPM 2010

Ahmad Almulhem, KFUPM 2010

ASCII Character Code

Ahmad Almulhem, KFUPM 2010

ASCII Character Code

capital vs smallA difference of (20)16 = 3210

Error Detecting Code

• In data communication, errors may happen• One code change into another code• How to detect errors?

• Add an extra bit called a parity bit such that• Number of 1’s is even (even parity) or odd (odd parity)

Ahmad Almulhem, KFUPM 2010

Ahmad Almulhem, KFUPM 2010

Error Detecting Code

ASCII A =ASCII T =

Conclusions

• Bits are bits• Modern digital devices represent everything as

collections of bits• A computer is one such digital device

• You can encode anything with sufficient 1’s and 0’s• Binary codes (BCD, gray code)• Text (ASCII)• Sound (.wav, .mp3, ...)• Pictures (.jpg, .gif, .tiff)

Ahmad Almulhem, KFUPM 2010